Precalculus Examples

$f (x) = x^{2} e^{x} - 81 e^{x}$

Step 1

Set $x^{2} e^{x} - 81 e^{x}$ equal to $0$ .

$x^{2} e^{x} - 81 e^{x} = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Factor $e^{x}$ out of $x^{2} e^{x} - 81 e^{x}$ .

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Step 2.1.1.1

Factor $e^{x}$ out of $x^{2} e^{x}$ .

$e^{x} x^{2} - 81 e^{x} = 0$

Step 2.1.1.2

Factor $e^{x}$ out of $- 81 e^{x}$ .

$e^{x} x^{2} + e^{x} \cdot - 81 = 0$

Step 2.1.1.3

Factor $e^{x}$ out of $e^{x} x^{2} + e^{x} \cdot - 81$ .

$e^{x} (x^{2} - 81) = 0$

Step 2.1.2

Rewrite $81$ as $9^{2}$ .

$e^{x} (x^{2} - 9^{2}) = 0$

Step 2.1.3

Factor.

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Step 2.1.3.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 9$ .

$e^{x} ((x + 9) (x - 9)) = 0$

Step 2.1.3.2

Remove unnecessary parentheses.

$e^{x} (x + 9) (x - 9) = 0$

Step 2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$e^{x} = 0$

$x + 9 = 0$

$x - 9 = 0$

Step 2.3

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 2.3.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 2.3.2

Solve $e^{x} = 0$ for $x$ .

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Step 2.3.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

$\ln (e^{x}) = \ln (0)$

Step 2.3.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 2.3.2.3

There is no solution for $e^{x} = 0$

No solution

Step 2.4

Set $x + 9$ equal to $0$ and solve for $x$ .

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Step 2.4.1

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

Step 2.4.2

Subtract $9$ from both sides of the equation.

$x = - 9$

Step 2.5

Set $x - 9$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Step 2.5.2

Add $9$ to both sides of the equation.

$x = 9$

Step 2.6

The final solution is all the values that make $e^{x} (x + 9) (x - 9) = 0$ true. The multiplicity of a root is the number of times the root appears.

$x = - 9$ (Multiplicity of $1$ )

$x = 9$ (Multiplicity of $1$ )

$x = - 9$ (Multiplicity of $1$ )

$x = 9$ (Multiplicity of $1$ )

Step 3